# Electric field fluctuations in QM optics

In a text I am using, it states the following: Given that $$\langle n| \hat{E}^2_x(z,t)| n \rangle = 2 \varepsilon_0^2 \sin^2(kz) \bigg( n + \frac{1}{2} \bigg)$$ The fluctuations in the electric field may be characterized by the variance $$\langle ( \Delta \hat{E}_x(z,t))^2 \rangle = \langle \hat{E}^2_x(z,t) \rangle - \langle \hat{E}_x(z,t) \rangle^2$$ or by the standard deviation $\Delta E_x = \langle ( \Delta E_x(z,t)^2)^{\frac{1}{2}}$, which is sometimes referred to as the uncertainty of the field. For the number state $|n \rangle$ we have $$\Delta E_x = \sqrt{2 \varepsilon_0} \sin (kz) \bigg( n + \frac{1}{2} \bigg)^\frac{1}{2}.$$ Note the even when $n = 0$, the field has fluctuations, the so-called vacuum fluctuations. Now the number states $|n \rangle$ are taken to represent a state of the field containing $n$ photons. Yet as we have seen, the average field is zero. This is all in accordance with the uncertainty principle because the number operator $\hat{n}$ does not commute with the electric field: $$[ \hat{n}, \hat{E}_x] = \varepsilon_0 \sin(kz) (\hat{a}^{\dagger}-\hat{a})$$ Thus $\hat{n}$ and $\hat{E}_x$ are complementary quantities for which their respective uncertainties obey the inequality $$\Delta n \Delta E_x \geq \frac{1}{2} \varepsilon_0 | \sin (kz)|| \langle \hat{a}^{\dagger} - \hat{a} \rangle |$$

For a number state $|n \rangle$, the right-hand side vanishes but $\Delta n = 0$ as well. If the field were accurately known, then the number of photons would be uncertain.

Questions: What is the importance of what is defined as fluctuations in the electric field? How does this operator notion of an electric field relate to the classical notion of the electric field as a wave? You simply go from a wave function to an operator seems a bit arbitrary?

Thanks.

• This means the fluctuation of the electric field is associated with the fluctuation in the photon number. – Lawrence B. Crowell Jul 7 '17 at 1:56
• @LawrenceB.Crowell Okay thanks. Do you maybe know what the significance of the observation "For a number state $|n \rangle$, the right-hand side vanishes but $\Delta n = 0$ as well" is? – user110903 Jul 10 '17 at 19:34